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a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)

Source: Czech-Polish-Slovak Junior Match 2015, Team p4 CPSJ

March 19, 2020
LCMGCDnumber theory

Problem Statement

Determine all such pairs pf positive integers (a,b)(a, b) such that a+b+(gcd(a,b))2=lcm(a,b)=2⋅lcm(a−1,b)a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b), where lcm(a,b)lcm (a, b) denotes the smallest common multiple, and gcd(a,b)gcd (a, b) denotes the greatest common divisor of numbers a,ba, b.
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